Derivation of the Integrated Rate Law for a First-Order Reaction

Derivation of the Integrated Rate Law for a First-Order Reaction

A first-order reaction is one where the rate of reaction depends linearly on the concentration of one reactant. The general form of a first-order reaction is:

$$A\to \text{Products}$$

The rate of the reaction can be expressed as:

$$\text{Rate}=-\frac{d[A]}{dt}=k[A]$$

where:

• $[A]$ is the concentration of the reactant $A$ at time $t$,
• $k$ is the rate constant of the reaction.

To derive the integrated rate law, we rearrange the equation to separate variables:

$$\frac{d[A]}{[A]}=-kdt$$

Now, integrate both sides. The left side is integrated with respect to $[A]$, and the right side with respect to $t$:

$$\int \frac{d[A]}{[A]}=-\int kdt$$

This gives:

$$\ln [A]=-kt+C$$

where $C$ is the integration constant. To determine $C$, we use the initial condition at $t=0$, where $[A]=[A{]}_0$ (the initial concentration of $A$):

$$\ln [A{]}_0=C$$

So, the equation becomes:

$$\ln [A]=-kt+\ln [A{]}_0$$

This can be rearranged to:

$$\ln \left(\frac{[A]}{[A{]}_0}\right)=-kt$$

Taking the exponential of both sides:

$$\frac{[A]}{[A{]}_0}=e^{-kt}$$

Finally, multiplying both sides by $[A{]}_0$:

$$[A]=[A{]}_0{e}^{-kt}$$

This is the integrated rate law for a first-order reaction. It shows that the concentration of the reactant decreases exponentially with time.

Significance of the Half-Life in First-Order Reactions

The half-life ($t_{1\mathrm{/}2}$) of a reaction is the time required for the concentration of the reactant to decrease to half of its initial value. For a first-order reaction, we can find the half-life by setting $[A]=\frac{[A{]}_0}{2}$ in the integrated rate law:

$$\frac{[A{]}_0}{2}=[A{]}_0{e}^{-k{t}_{1\mathrm{/}2}}$$

Dividing both sides by $[A{]}_0$ gives:

$$\frac{1}{2}=e^{-k{t}_{1\mathrm{/}2}}$$

Taking the natural logarithm of both sides:

$$\ln \left(\frac{1}{2}\right)=-k{t}_{1\mathrm{/}2}$$

Since $\ln \left(\frac{1}{2}\right)=-\ln 2$:

$$-\ln 2=-k{t}_{1\mathrm{/}2}$$$$t_{1\mathrm{/}2}=\frac{\ln 2}{k}$$

The half-life for a first-order reaction is given by:

$$t_{1\mathrm{/}2}=\frac{0.693}{k}$$

Importance of Half-Life:

1. Constant Half-Life: In first-order reactions, the half-life is constant and does not depend on the initial concentration of the reactant. This is a key characteristic of first-order kinetics.

2. Practical Use: The half-life is a useful measure in various applications, such as in pharmacokinetics, where it helps determine how long a drug will stay active in the body.

3. Reaction Monitoring: Knowing the half-life allows chemists to predict how long it will take for a reactant to be consumed to a certain extent, which is useful in controlling and optimizing reactions in industrial processes.

In summary, the integrated rate law for a first-order reaction describes the exponential decay of the reactant concentration over time, and the half-life provides a constant time measure for the reaction's progress, irrespective of the initial concentration.